cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A161804 G.f.: A(q) = exp( Sum_{n>=1} A002129(n) * 3*A038500(n) * q^n/n ).

Original entry on oeis.org

1, 3, 3, 12, 30, 27, 66, 141, 111, 255, 513, 378, 903, 1815, 1356, 2970, 5727, 4131, 8571, 15882, 10881, 23001, 42417, 29106, 59763, 108165, 73500, 145164, 255831, 167643, 333693, 585258, 382053, 751059, 1302966, 849339, 1623009, 2762349
Offset: 0

Views

Author

Paul D. Hanna, Jul 20 2009

Keywords

Comments

A002129 forms the l.g.f. of log[ Sum_{n>=0} x^(n(n+1)/2) ], while 3*A038500 forms the l.g.f. of the log of the g.f. of A161809 and A038500(n) is the highest power of 3 dividing n.

Examples

			G.f.: A(q) = 1 + 3*q + 3*q^2 + 12*q^3 + 30*q^4 + 27*q^5 + 66*q^6 + ...
log(A(q)) = 3*q - 3*q^2 + 36*q^3 - 15*q^4 + 18*q^5 - 36*q^6 + 24*q^7 + ...
Sum_{n>=1} A002129(n)*q^n/n = log(1 + q + q^3 + q^6 + q^10 + q^15 + ...),
Sum_{n>=1} 3*A038500(n)*x^n/n = log of the g.f. of A161809.
TRISECTIONS:
T_0(q) = 1 + 12*q + 66*q^2 + 255*q^3 + 903*q^4 + 2970*q^5 + ... (A161805)
T_1(q) = 3 + 30*q + 141*q^2 + 513*q^3 + 1815*q^4 + 5727*q^5 + ... (A161806)
T_2(q) = 3 + 27*q + 111*q^2 + 378*q^3 + 1356*q^4 + 4131*q^5 + ... (A161807)
where T_1(-q)/T_0(-q)/3 equals (cf. A132977):
1 + 2*q + 5*q^2 + 12*q^3 + 26*q^4 + 50*q^5 + 92*q^6 + 168*q^7 + ...
and T_2(-q)/T_0(-q)/3 equals (cf. A132978):
1 + 3*q + 7*q^2 + 15*q^3 + 32*q^4 + 63*q^5 + 114*q^6 + 201*q^7 + ...
also, T_2(q)/T_1(q) equals (cf. A092848):
1 - q + 2*q^3 - 2*q^4 - q^5 + 4*q^6 - 4*q^7 - q^8 + 8*q^9 - 8*q^10 + ...

Crossrefs

Cf. trisections: A161805 (T_0), A161806 (T_1), A161807 (T_2).
Cf. A132977 (T_1/T_0), A132978 (T_2/T_0), A092848 (T_2/T_1).
Cf. A002129, A038500, A161809, A161800 (variant).

Programs

  • PARI
    {a(n)=local(L=sum(m=1, n,3*3^valuation(m,3)*sumdiv(m, d, -(-1)^d*d)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

Formula

Given trisections where A(q) = T_0(q^3) + q*T_1(q^3) + q^2*T_2(q^3):
T_0(q) = Sum_{n>=0} a(3n)*q^n,
T_1(q) = Sum_{n>=0} a(3n+1)*q^n,
T_2(q) = Sum_{n>=0} a(3n+2)*q^n,
then it appears that:
T_1(-q)/T_0(-q) = 3*q^(-1/3)*(eta(q^6)^4/(eta(q)*eta(q^3)*eta(q^4)*eta(q^12)))^2 (Cf. A132977);
T_2(-q)/T_0(-q) = 3*q^(-2/3)*(eta(q^2)*eta(q^6))^2*eta(q^3)*eta(q^12)/(eta(q)*eta(q^4))^3 (cf. A132978);
T_2(q)/T_1(q) = g.f. of A092848, the reciprocal of Hauptmodul for Gamma_0(18).

A182000 G.f.: exp( Sum_{n>=1} 2^A090740(n) * x^n/n ) where A090740(n) = highest exponent of 2 in 3^n-1.

Original entry on oeis.org

1, 2, 6, 10, 22, 34, 62, 90, 150, 210, 326, 442, 654, 866, 1230, 1594, 2198, 2802, 3766, 4730, 6230, 7730, 9998, 12266, 15630, 18994, 23878, 28762, 35742, 42722, 52526, 62330, 75926, 89522, 108118, 126714, 151878, 177042, 210702, 244362, 288982, 333602, 392182
Offset: 0

Views

Author

Paul D. Hanna, Apr 17 2012

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 6*x^2 + 10*x^3 + 22*x^4 + 34*x^5 + 62*x^6 +...
The g.f. satisfies:
A(x)/A(x^2) = 1 + 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 +...+ 2*n*x^n +...
The logarithm of the g.f. begins:
log(A(x)) = 2*x + 8*x^2/2 + 2*x^3/3 + 16*x^4/4 + 2*x^5/5 + 8*x^6/6 + 2*x^7/7 + 32*x^8/8 + 2*x^9/9 + 8*x^10/10 + 2*x^11/11 + 16*x^12/12 +...+ 2^A090740(n)*x^n/n +...
where the highest exponents of 2 in 3^n-1, for n>=1, begins:
A090740 = [1,3,1,4,1,3,1,5,1,3,1,4,1,3,1,6,1,3,1,4,1,3,1,5,1,3,1,4,1,...].
The g.f.s of the BISECTIONS begin:
B_0(x) = 1 + 6*x + 22*x^2 + 62*x^3 + 150*x^4 + 326*x^5 + 654*x^6 +...
B_1(x) = 2 + 10*x + 34*x^2 + 90*x^3 + 210*x^4 + 442*x^5 + 866*x^6 +...
where 2 * B_0(x) / B_1(x) = 1+x.

Links

Crossrefs

Cf. A173283, A161809, A182185, A090740.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n+1,2^valuation(3^m-1,2)*x^m/m)+x*O(x^n)),n)}
for(n=0,40,print1(a(n),", "))
  • PARI
    {a(n)=local(A=1+x);for(i=1,#binary(n)+1,A=(1+x^2)/(1-x)^2*subst(A,x,x^2+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = (1+x^2)/(1-x)^2 * A(x^2).
Define BISECTIONS: A(x) = B_0(x^2) + x*B_1(x^2), then: B_1(x)/B_0(x) = 2/(1+x).

A183038 G.f.: exp( Sum_{n>=1} A051064(n)*3^A051064(n)*x^n/n ) where A051064(n) equals the 3-adic valuation of 3n.

Original entry on oeis.org

1, 3, 6, 15, 30, 51, 93, 156, 240, 387, 597, 870, 1311, 1920, 2697, 3873, 5448, 7422, 10278, 14016, 18636, 25098, 33402, 43548, 57333, 74757, 95820, 123780, 158637, 200391, 254778, 321798, 401451, 503490, 627915, 774726, 960156, 1184205, 1446873
Offset: 0

Views

Author

Paul D. Hanna, Dec 19 2010

Keywords

Comments

Compare to B(x), the g.f. of the number of partitions of 3n into powers of 3 (A005704):
B(x) = exp( Sum_{n>=1} 3^A051064(n)*x^n/n ) = (1-x)^(-1)*Product_{n>=0} 1/(1 - x^(3^n)).

Examples

			G.f.: A(x) =  1 + 3*x + 6*x^2 + 15*x^3 + 30*x^4 + 51*x^5 + 93*x^6 +...
log(A(x)) = 3*x + 3*x^2/2 + 18*x^3/3 + 3*x^4/4 + 3*x^5/5 + 18*x^6/6 + 3*x^7/7 + 3*x^8/8 + 81*x^9/9 + 3*x^10/10 + 3*x^11/11 + 18*x^12/12 +...
G.f. satisfies A(x) = A(x^3)*G(x) where G(x) is the g.f. of A161809:
G(x) = 1 + 3*x + 6*x^2 + 12*x^3 + 21*x^4 + 33*x^5 + 51*x^6 +...
TRISECTIONS of g.f. begin:
T_0(x) = 1 + 15*x + 93*x^2 + 387*x^3 + 1311*x^4 + 3873*x^5 +...
T_1(x) = 3 + 30*x + 156*x^2 + 597*x^3 + 1920*x^4 + 5448*x^5 +...
T_2(x) = 6 + 51*x + 240*x^2 + 870*x^3 + 2697*x^4 + 7422*x^5 +...
where the ratios involve Fibonacci numbers:
T_1(x)/T_0(x) = 3*(1 - 5*x + 34*x^2 - 233*x^3 +...+ (-1)^n*Fibonacci(4n+1)*x^n +...);
T_2(x)/T_0(x) = 3*(2 - 13*x + 89*x^2 - 610*x^3 +...+ (-1)^n*Fibonacci(4n+3)*x^n +...).

Crossrefs

Cf. A161809, A183039, A051064, A005704, A183036.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1,n,valuation(3*m,3)*3^valuation(3*m,3)*x^m/m)+x*O(x^n)),n)}

Formula

G.f. satisfies: A(x) = (1-x^3)/(1-x)^3 * A(x^3)^2/A(x^9).
G.f. satisfies: A(x) = A(x^3)*G(x) where G(x) = G(x^3)*(1+x+x^2)/(1-x)^2 is the g.f. of A161809.
Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:
T_1(x)/T_0(x) = 3*(1 + 2*x)/(1 + 7*x + x^2) and
T_2(x)/T_0(x) = 3*(2 + x)/(1 + 7*x + x^2).

A182185 G.f.: exp( Sum_{n>=1} 3^b(n) * x^n/n ) where b(n) = highest exponent of 3 in 2^n+1.

Original entry on oeis.org

1, 3, 5, 9, 15, 21, 29, 39, 49, 63, 81, 99, 123, 153, 183, 219, 261, 303, 353, 411, 469, 537, 615, 693, 781, 879, 977, 1089, 1215, 1341, 1485, 1647, 1809, 1989, 2187, 2385, 2607, 2853, 3099, 3375, 3681, 3987, 4323, 4689, 5055, 5457, 5895, 6333, 6813, 7335, 7857, 8421
Offset: 0

Views

Author

Paul D. Hanna, Apr 18 2012

Keywords

Examples

			G.f.: A(x) = 1 + 3*x + 5*x^2 + 9*x^3 + 15*x^4 + 21*x^5 + 29*x^6 + 39*x^7 +...
The g.f. satisfies:
A(x)/A(x^3) = 1 + 3*x + 5*x^2 + 6*x^3 + 6*x^4 + 6*x^5 +...+ 6*x^n +...
The logarithm of the g.f. begins:
log(A(x)) = 3*x + x^2/2 + 9*x^3/3 + x^4/4 + 3*x^5/5 + x^6/6 + 3*x^7/7 + x^8/8 + 27*x^9/9 + x^10/10 + 3*x^11/11 + x^12/12 +...+ 3^b(n)*x^n/n +...
where b(n) = highest exponent of 3 in 2^n+1, for n>=1, and begins:
b = [1,0,2,0,1,0,1,0,3,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,4,...].
The g.f.s of the TRISECTIONS begin:
T_0(x) = 1 + 9*x + 29*x^2 + 63*x^3 + 123*x^4 + 219*x^5 + 353*x^6 +...
T_1(x) = 3 + 15*x + 39*x^2 + 81*x^3 + 153*x^4 + 261*x^5 + 411*x^6 +...
T_2(x) = 5 + 21*x + 49*x^2 + 99*x^3 + 183*x^4 + 303*x^5 + 469*x^6 +...
where T_1(x)/T_0(x) = 3*(1+x)/(1+5*x), T_2(x)/T_0(x) = (5+x)/(1+5*x).

Crossrefs

Cf. A182000, A161809.

Programs

  • PARI
    {a(n)=polcoeff(exp(sum(m=1, n+1, 3^valuation(2^m+1, 3)*x^m/m)+x*O(x^n)), n)}
for(n=0, 65, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1+x);for(i=1,ceil(log(n+1)/log(3)),A=(1-x^2)*(1-x^3)/(1-x)^3*subst(A,x,x^3+x*O(x^n)));polcoeff(A,n)}

Formula

G.f. satisfies: A(x) = (1-x^2)*(1-x^3)/(1-x)^3 * A(x^3).
Define TRISECTIONS: A(x) = T_0(x^3) + x*T_1(x^3) + x^2*T_2(x^3), then:
(1) T_1(x)/T_0(x) = 3*(1+x)/(1+5*x),
(2) T_2(x)/T_0(x) = (5+x)/(1+5*x),
(3) T_0(x)/T_0(x^3) = (1+x)*(1+5*x)*(1-x^3)^2 / ((1-x)^3*(1+5*x^3)),
(4) T_1(x)/T_1(x^3) = (1+x)^2*(1-x^3)^2 / ((1-x)^3*(1+x^3)),
(5) T_2(x)/T_2(x^3) = (1+x)*(5+x)*(1-x^3)^2 / ((1-x)^3*(5+x^3)),
(6) A(x) = (1-x)/(1+5*x)*T_0(x) = (1-x)/(1+x)*T_1(x)/3 = (1-x)/(5+x)*T_2(x).

A195760 G.f.: A(x) = exp( Sum_{n>=1} 5*5^A112765(n) * x^n/n ), where A112765 is the exponent of the highest power of 5 dividing n.

Original entry on oeis.org

1, 5, 15, 35, 70, 130, 230, 390, 635, 995, 1515, 2255, 3290, 4710, 6620, 9160, 12505, 16865, 22485, 29645, 38695, 50055, 64215, 81735, 103245, 129505, 161405, 199965, 246335, 301795, 367855, 446255, 538965, 648185, 776345, 926265, 1101155, 1304615, 1540635
Offset: 0

Views

Author

Paul D. Hanna, Sep 23 2011

Keywords

Examples

			G.f.: A(x) = 1 + 5*x + 15*x^2 + 35*x^3 + 70*x^4 + 130*x^5 + 230*x^6 +...
log(A(x)) = 5*x + 5*x^2/2 + 5*x^3/3 + 5*x^4/4 + 25*x^5/5 + 5*x^6/6 + 5*x^7/7 + 5*x^8/8 + 5*x^9/9 + 25*x^10/10 +...
The coefficients in the QUINTISECTIONS of g.f. A(x) begin:
Q0: [1, 130, 1515, 9160, 38695, 129505, 367855, 926265, 2128510, ...];
Q1: [5, 230, 2255, 12505, 50055, 161405, 446255, 1101155, 2491030, ...];
Q2: [15, 390, 3290, 16865, 64215, 199965, 538965, 1304615, 2907440, ...];
Q3: [35, 635, 4710, 22485, 81735, 246335, 648185, 1540635, 3384660, ...];
Q4: [70, 995, 6620, 29645, 103245, 301795, 776345, 1813595, 3930245, ...].
The coefficients in the products Q2*Q3 and Q1*Q4 begin:
Q2(x)*Q3(x): [525, 23175, 433450, 4853600, 38447875, 236756775, ...];
Q1(x)*Q4(x): [350, 21075, 419800, 4789900, 38209000, 235990975, ...];
where Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2, and
R(x) = 1 - 9*x + 36*x^2 - 84*x^3 + 126*x^4 - 130*x^5 + 120*x^6 +...

Links

Crossrefs

Cf. A195761, A161809, A112765.

Programs

  • PARI
    {a(n)=local(N=ceil(log(n+6)/log(5)));polcoeff(1/(1-x+x*O(x^n))^5/prod(k=1,N,(1-x^(5^k) +x*O(x^n))^4),n)}
  • PARI
    {a(n)=local(L=sum(m=1, n, 5*5^valuation(m, 5)*x^m/m)+x*O(x^n)); polcoeff(exp(L), n)}

Formula

G.f.: A(x) = 1/(1-x)^5 * Product_{n>=1} 1/(1 - x^(5^n))^4.
G.f. satisfies: A(x) = A(x^5)*(1-x^5)/(1-x)^5.
Let the QUINTISECTIONS of g.f. A(x) be defined by:
A(x) = Q0(x^5) + x*Q1(x^5) + x^2*Q3(x^5) + x^3*Q3(x^5) + x^4*Q4(x^5),
then:
_ Q0(x) = (1 + 121*x + 381*x^2 + 121*x^3 + x^4)/R(x)
_ Q1(x) = 5*(1 + 37*x + 73*x^2 + 14*x^3)/R(x)
_ Q2(x) = 5*(3 + 51*x + 64*x^2 + 7*x^3)/R(x)
_ Q3(x) = 5*(7 + 64*x + 51*x^2 + 3*x^3)/R(x)
_ Q4(x) = 5*(14 + 73*x + 37*x^2 + 1*x^3)/R(x)
where R(x) = (1-x)^5 * Product_{n>=0} (1 - x^(5^n))^4.
Further, the quintisections are related by:
_ Q2(x)*Q3(x) - Q1(x)*Q4(x) = 175*(1-x)^6/R(x)^2.

A377555 E.g.f.: exp(Sum_{n>=1} A038500(n) * x^n).

Original entry on oeis.org

1, 1, 3, 25, 121, 861, 10051, 88453, 972945, 16663321, 205667011, 3069838641, 61038456073, 997387656565, 18623707785411, 426663334715101, 8606752819074721, 192052302116929713, 5139946157328092035, 122142504609497184841, 3172736666738570349081, 94751480557190553846541
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2024

Keywords

Crossrefs

Cf. A038500, A161809, A377556.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sum[3^IntegerExponent[k, 3]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0,nmax]!

A377556 E.g.f.: exp(Sum_{n>=1} A006519(n) * x^n).

Original entry on oeis.org

1, 1, 5, 19, 193, 1181, 13021, 117895, 1868609, 20980153, 348219541, 4940639771, 98898110785, 1632238421269, 34910480911853, 672959412044431, 16733065940227201, 359936040496423025, 9469928134781142949, 229631546862609396643, 6716832478519734558401, 178344294076141938008461
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2024

Keywords

Crossrefs

Cf. A000123, A006519, A038500, A161809, A377555.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[Sum[2^IntegerExponent[k, 2]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0,nmax]!
Showing 1-7 of 7 results.

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